Single-variable calculus from first principles
Informally, a function is continuous if you can draw it without lifting your pen: no holes, no jumps, no sudden blow-ups. The precise version pins this down with the limit you just learned: at every point, where the function is heading must match where it actually is.
Three things must all line up: f(a) exists, the limit exists, and they're equal. If any of the three fails, you have a discontinuity, and there are exactly three kinds.
A removable discontinuity is a single missing point, a hole, where the limit exists but the function skipped that value (like the (x²−4)/(x−2) hole). A jump is when the left and right limits disagree, so the graph leaps from one level to another. An infinite discontinuity is a vertical asymptote, where the function shoots off to ±∞ (like 1/x at 0).